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<h1 class="libtitle">Library mathcomp.ssreflect.finfun</h1>
<div class="code">
<span class="comment">(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. <br/>
Distributed under the terms of CeCILL-B. *)</span><br/>
<br/>
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<div class="doc">
This file implements a type for functions with a finite domain:
{ffun aT -> rT} where aT should have a finType structure,
{ffun forall x : aT, rT} for dependent functions over a finType aT,
and {ffun funT} where funT expands to a product over a finType.
Any eqType, choiceType, countType and finType structures on rT extend to
{ffun aT -> rT} as Leibnitz equality and extensional equalities coincide.
(T ^ n)%type is notation for {ffun 'I_n -> T}, which is isomorphic
to n.-tuple T, but is structurally positive and thus can be used to
define inductive types, e.g., Inductive tree := node n of tree ^ n (see
mid-file for an expanded example).
> More generally, {ffun fT} is always structurally positive.
{ffun fT} inherits combinatorial structures of rT, i.e., eqType,
choiceType, countType, and finType. However, due to some limitations of
the Coq 8.9 unification code the structures are only inherited in the
NON dependent case, when rT does not depend on x.
For f : {ffun fT} with fT := forall x : aT, rT we define
f x == the image of x under f (f coerces to a CiC function)
> The coercion is structurally decreasing, e.g., Coq will accept
Fixpoint size t := let: node n f := t in sumn (codom (size \o f)) + 1.
as structurally decreasing on t of the inductive tree type above.
{dffun fT} == alias for {ffun fT} that inherits combinatorial
structures on rT, when rT DOES depend on x.
total_fun g == the function induced by a dependent function g of type
forall x, rT on the total space {x : aT & rT}.
:= fun x => Tagged (fun x => rT) (g x).
tfgraph f == the total function graph of f, i.e., the #|aT|.-tuple
of all the (dependent pair) values of total_fun f.
finfun g == the f extensionally equal to g, and the RECOMMENDED
interface for building elements of {ffun fT}.
[ffun x : aT => E] := finfun (fun x : aT => E).
There should be an explicit type constraint on E if
type does not depend on x, due to the Coq unification
limitations referred to above.
ffun0 aT0 == the trivial finfun, from a proof aT0 that #|aT| = 0.
f \in family F == f belongs to the family F (f x \in F x for all x)
There are addidional operations for non-dependent finite functions,
i.e., f in {ffun aT -> rT}.
[ffun x => E] := finfun (fun x => E).
The type of E must not depend on x; this restriction
is a mitigation of the aforementioned Coq unification
limitations.
[ffun=> E] := [ffun _ => E] (E should not have a dependent type).
fgraph f == the function graph of f, i.e., the #|aT|.-tuple
listing the values of f x, for x ranging over enum aT.
Finfun G == the finfun f whose (simple) function graph is G.
f \in ffun_on R == the range of f is a subset of R.
y.-support f == the y-support of f, i.e., [pred x | f x != y].
Thus, y.-support f \subset D means f has y-support D.
We will put Notation support := 0.-support in ssralg.
f \in pffun_on y D R == f is a y-partial function from D to R:
f has y-support D and f x \in R for all x \in D.
f \in pfamily y D F == f belongs to the y-partial family from D to F:
f has y-support D and f x \in F x for all x \in D.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="Def"><span class="id" title="section">Def</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variables</span> (<a name="Def.aT"><span class="id" title="variable">aT</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<a name="Def.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.ssreflect.finfun.html#aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>).<br/>
<br/>
<span class="id" title="keyword">Inductive</span> <a name="finfun_on"><span class="id" title="inductive">finfun_on</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#Def.aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/>
| <a name="finfun_nil"><span class="id" title="constructor">finfun_nil</span></a> : <a class="idref" href="mathcomp.ssreflect.finfun.html#finfun_on"><span class="id" title="inductive">finfun_on</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a><br/>
| <a name="finfun_cons"><span class="id" title="constructor">finfun_cons</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.ssreflect.finfun.html#Def.rT"><span class="id" title="variable">rT</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a> & <a class="idref" href="mathcomp.ssreflect.finfun.html#finfun_on"><span class="id" title="inductive">finfun_on</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#s"><span class="id" title="variable">s</span></a> : <a class="idref" href="mathcomp.ssreflect.finfun.html#finfun_on"><span class="id" title="inductive">finfun_on</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#s"><span class="id" title="variable">s</span></a>).<br/>
<br/>
<br/>
<br/>
<span class="id" title="keyword">Variant</span> <a name="finfun_of"><span class="id" title="inductive">finfun_of</span></a> (<span class="id" title="var">ph</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#phant"><span class="id" title="inductive">phant</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.finfun.html#Def.rT"><span class="id" title="variable">rT</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a>)) : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#predArgType"><span class="id" title="definition">predArgType</span></a> :=<br/>
<a name="FinfunOf"><span class="id" title="constructor">FinfunOf</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.ssreflect.finfun.html#finfun_on"><span class="id" title="inductive">finfun_on</span></a> (<a class="idref" href="mathcomp.ssreflect.fintype.html#enum"><span class="id" title="abbreviation">enum</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#Def.aT"><span class="id" title="variable">aT</span></a>).<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="dfinfun_of"><span class="id" title="definition">dfinfun_of</span></a> <span class="id" title="var">ph</span> := <a class="idref" href="mathcomp.ssreflect.finfun.html#finfun_of"><span class="id" title="inductive">finfun_of</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#ph"><span class="id" title="variable">ph</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="fun_of_fin"><span class="id" title="definition">fun_of_fin</span></a> <span class="id" title="var">ph</span> (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.ssreflect.finfun.html#finfun_of"><span class="id" title="inductive">finfun_of</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#ph"><span class="id" title="variable">ph</span></a>) <span class="id" title="var">x</span> :=<br/>
<span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.ssreflect.finfun.html#FinfunOf"><span class="id" title="constructor">FinfunOf</span></a> <span class="id" title="var">f_aT</span> := <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.finfun.html#fun_of_fin_rec"><span class="id" title="definition">fun_of_fin_rec</span></a> <span class="id" title="var">f_aT</span> (<a class="idref" href="mathcomp.ssreflect.fintype.html#mem_enum"><span class="id" title="lemma">mem_enum</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#Def.aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a>).<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.finfun.html#Def"><span class="id" title="section">Def</span></a>.<br/>
<br/>
<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.ssreflect.finfun.html#fun_of_fin"><span class="id" title="definition">fun_of_fin</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#fun_of_fin"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#fun_of_fin"><span class="id" title="definition">finfun_of</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#fun_of_fin"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#fun_of_fin"><span class="id" title="definition">Funclass</span></a>.<br/>
<span class="id" title="keyword">Identity</span> <span class="id" title="keyword">Coercion</span> <span class="id" title="var">unfold_dfinfun_of</span> : <span class="id" title="var">dfinfun_of</span> >-> <span class="id" title="var">finfun_of</span>.<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">"</span></a>{ 'ffun' fT }" := (<a class="idref" href="mathcomp.ssreflect.finfun.html#finfun_of"><span class="id" title="inductive">finfun_of</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#Phant"><span class="id" title="constructor">Phant</span></a> <span class="id" title="var">fT</span>))<br/>
(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "{ 'ffun' '[hv' fT ']' }") : <span class="id" title="var">type_scope</span>.<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="15e3a46a8a60865a7b5408d87034cd4e"><span class="id" title="notation">"</span></a>{ 'dffun' fT }" := (<a class="idref" href="mathcomp.ssreflect.finfun.html#dfinfun_of"><span class="id" title="definition">dfinfun_of</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#Phant"><span class="id" title="constructor">Phant</span></a> <span class="id" title="var">fT</span>))<br/>
(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "{ 'dffun' '[hv' fT ']' }") : <span class="id" title="var">type_scope</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="exp_finIndexType"><span class="id" title="definition">exp_finIndexType</span></a> := <a class="idref" href="mathcomp.ssreflect.fintype.html#ordinal_finType"><span class="id" title="definition">ordinal_finType</span></a>.<br/>
<span class="id" title="keyword">Notation</span> <a name="c93a2e1bb8503fc4a9598804b268d1be"><span class="id" title="notation">"</span></a>T ^ n" :=<br/>
(@<a class="idref" href="mathcomp.ssreflect.finfun.html#finfun_of"><span class="id" title="inductive">finfun_of</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#exp_finIndexType"><span class="id" title="definition">exp_finIndexType</span></a> <span class="id" title="var">n</span>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#94502d422d70bbaf4f390946d9885b7c"><span class="id" title="notation">fun</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#94502d422d70bbaf4f390946d9885b7c"><span class="id" title="notation">⇒</span></a> <span class="id" title="var">T</span>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#Phant"><span class="id" title="constructor">Phant</span></a> <span class="id" title="var">_</span>)) : <span class="id" title="var">type_scope</span>.<br/>
<br/>
<br/>
<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a name="FinfunDefSig"><span class="id" title="module">FinfunDefSig</span></a>.<br/>
<span class="id" title="keyword">Parameter</span> <a name="FinfunDefSig.finfun"><span class="id" title="axiom">finfun</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">aT</span> <span class="id" title="var">rT</span>, <a class="idref" href="mathcomp.ssreflect.finfun.html#finPi"><span class="id" title="abbreviation">finPi</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#rT"><span class="id" title="variable">rT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">ffun</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#finPi"><span class="id" title="abbreviation">finPi</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">}</span></a>.<br/>
<span class="id" title="keyword">Axiom</span> <a name="FinfunDefSig.finfunE"><span class="id" title="axiom">finfunE</span></a> : <a class="idref" href="mathcomp.ssreflect.finfun.html#FinfunDefSig.finfun"><span class="id" title="axiom">finfun</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#finfun_def"><span class="id" title="abbreviation">finfun_def</span></a>.<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.finfun.html#FinfunDefSig"><span class="id" title="module">FinfunDefSig</span></a>.<br/>
<br/>
<span class="id" title="keyword">Module</span> <a name="FinfunDef"><span class="id" title="module">FinfunDef</span></a> : <a class="idref" href="mathcomp.ssreflect.finfun.html#FinfunDefSig"><span class="id" title="module">FinfunDefSig</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="FinfunDef.finfun"><span class="id" title="definition">finfun</span></a> := <a class="idref" href="mathcomp.ssreflect.finfun.html#finfun_def"><span class="id" title="abbreviation">finfun_def</span></a>.<br/>
<span class="id" title="keyword">Lemma</span> <a name="FinfunDef.finfunE"><span class="id" title="lemma">finfunE</span></a> : <a class="idref" href="mathcomp.ssreflect.finfun.html#FinfunDef.finfun"><span class="id" title="definition">finfun</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#finfun_def"><span class="id" title="abbreviation">finfun_def</span></a>. <br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.finfun.html#FinfunDef"><span class="id" title="module">FinfunDef</span></a>.<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="finfun"><span class="id" title="abbreviation">finfun</span></a> := <a class="idref" href="mathcomp.ssreflect.finfun.html#finfun"><span class="id" title="axiom">FinfunDef.finfun</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">finfun_unlock</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#Unlockable"><span class="id" title="constructor">Unlockable</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#finfunE"><span class="id" title="axiom">FinfunDef.finfunE</span></a>.<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="e4e2ffb93b77700f7a723d1db6d75bdf"><span class="id" title="notation">"</span></a>[ 'ffun' x : aT => E ]" := (<a class="idref" href="mathcomp.ssreflect.finfun.html#finfun"><span class="id" title="abbreviation">finfun</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> : <span class="id" title="var">aT</span> ⇒ <span class="id" title="var">E</span>))<br/>
(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">x</span> <span class="id" title="var">ident</span>) : <span class="id" title="var">fun_scope</span>.<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="486743bb05c6aa8b9d64fd3cec29ee79"><span class="id" title="notation">"</span></a>[ 'ffun' x => E ]" := (@<a class="idref" href="mathcomp.ssreflect.finfun.html#finfun"><span class="id" title="abbreviation">finfun</span></a> <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#94502d422d70bbaf4f390946d9885b7c"><span class="id" title="notation">fun</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#94502d422d70bbaf4f390946d9885b7c"><span class="id" title="notation">⇒</span></a> <span class="id" title="var">_</span>) (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <span class="id" title="var">E</span>))<br/>
(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">x</span> <span class="id" title="var">ident</span>, <span class="id" title="var">format</span> "[ 'ffun' x => E ]") : <span class="id" title="var">fun_scope</span>.<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="765311140842c5fd14103e5433ef110e"><span class="id" title="notation">"</span></a>[ 'ffun' => E ]" := <a class="idref" href="mathcomp.ssreflect.finfun.html#486743bb05c6aa8b9d64fd3cec29ee79"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#486743bb05c6aa8b9d64fd3cec29ee79"><span class="id" title="notation">ffun</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.ssreflect.finfun.html#486743bb05c6aa8b9d64fd3cec29ee79"><span class="id" title="notation">⇒</span></a> <span class="id" title="var">E</span><a class="idref" href="mathcomp.ssreflect.finfun.html#486743bb05c6aa8b9d64fd3cec29ee79"><span class="id" title="notation">]</span></a><br/>
(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'ffun' => E ]") : <span class="id" title="var">fun_scope</span>.<br/>
<br/>
<span class="comment">(* Example outcommented.<br/>
<span class="comment">(** Examples of using finite functions as containers in recursive inductive <br/>
types, and making use of the fact that the type and accessor are <br/>
structurally positive and decreasing, respectively. **)</span><br/>
Unset Elimination Schemes.<br/>
Inductive tree := node n of tree ^ n.<br/>
Fixpoint size t := let: node n f := t in sumn (codom (size \o f)) + 1.<br/>
Example tree_step (K : tree -> Type) :=<br/>
forall n st (t := node st) & forall i : 'I_n, K (st i), K t.<br/>
Example tree_rect K (Kstep : tree_step K) : forall t, K t.<br/>
Proof. by fix IHt 1 => -<span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode"><span class="id" title="var">st</span></span>; apply/Kstep=> i; apply: IHt. Defined.<br/>
<br/>
<span class="comment">(** An artificial example use of dependent functions. **)</span><br/>
Inductive tri_tree n := tri_row of {ffun forall i : 'I_n, tri_tree i}.<br/>
Fixpoint tri_size n (t : tri_tree n) :=<br/>
let: tri_row f := t in sumn <span class="inlinecode"><span class="id" title="var">seq</span></span> <span class="inlinecode"><span class="id" title="var">tri_size</span></span> <span class="inlinecode">(<span class="id" title="var">f</span></span> <span class="inlinecode"><span class="id" title="var">i</span>)</span> <span class="inlinecode">|</span> <span class="inlinecode"><span class="id" title="var">i</span></span> <span class="inlinecode">:</span> <span class="inlinecode">'<span class="id" title="var">I_n</span></span> + 1.<br/>
Example tri_tree_step (K : forall n, tri_tree n -> Type) :=<br/>
forall n st (t := tri_row st) & forall i : 'I_n, K i (st i), K n t.<br/>
Example tri_tree_rect K (Kstep : tri_tree_step K) : forall n t, K n t.<br/>
Proof. by fix IHt 2 => n <span class="inlinecode"><span class="id" title="var">st</span></span>; apply/Kstep=> i; apply: IHt. Defined.<br/>
Set Elimination Schemes.<br/>
<span class="comment">(** End example. *)</span> **)</span><br/>
<br/>
</div>
<div class="doc">
The correspondance between finfun_of and CiC dependent functions.
</div>
<div class="code">
<span class="id" title="keyword">Section</span> <a name="DepPlainTheory"><span class="id" title="section">DepPlainTheory</span></a>.<br/>
<span class="id" title="keyword">Variables</span> (<a name="DepPlainTheory.aT"><span class="id" title="variable">aT</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<a name="DepPlainTheory.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.ssreflect.finfun.html#aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>).<br/>
<span class="id" title="keyword">Notation</span> <a name="fT"><span class="id" title="abbreviation">fT</span></a> := <a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">ffun</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#finPi"><span class="id" title="abbreviation">finPi</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#DepPlainTheory.aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#DepPlainTheory.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">}</span></a>.<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">f</span> : <a class="idref" href="mathcomp.ssreflect.finfun.html#fT"><span class="id" title="abbreviation">fT</span></a>.<br/>
<br/>
<span class="id" title="keyword">Fact</span> <a name="ffun0"><span class="id" title="lemma">ffun0</span></a> (<span class="id" title="var">aT0</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#DepPlainTheory.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0) : <a class="idref" href="mathcomp.ssreflect.finfun.html#fT"><span class="id" title="abbreviation">fT</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="ffunE"><span class="id" title="lemma">ffunE</span></a> <span class="id" title="var">g</span> <span class="id" title="var">x</span> : (<a class="idref" href="mathcomp.ssreflect.finfun.html#finfun"><span class="id" title="abbreviation">finfun</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#fT"><span class="id" title="abbreviation">fT</span></a>) <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="ffunP"><span class="id" title="lemma">ffunP</span></a> (<span class="id" title="var">f1</span> <span class="id" title="var">f2</span> : <a class="idref" href="mathcomp.ssreflect.finfun.html#fT"><span class="id" title="abbreviation">fT</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#4bfb4f2d0721ba668e3a802ab1b745a1"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.finfun.html#f1"><span class="id" title="variable">f1</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f2"><span class="id" title="variable">f2</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#4bfb4f2d0721ba668e3a802ab1b745a1"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#4bfb4f2d0721ba668e3a802ab1b745a1"><span class="id" title="notation">↔</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f1"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f2"><span class="id" title="variable">f2</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="ffunK"><span class="id" title="lemma">ffunK</span></a> : @<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#finPi"><span class="id" title="abbreviation">finPi</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#DepPlainTheory.aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#DepPlainTheory.rT"><span class="id" title="variable">rT</span></a>) <a class="idref" href="mathcomp.ssreflect.finfun.html#fT"><span class="id" title="abbreviation">fT</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#fun_of_fin"><span class="id" title="definition">fun_of_fin</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#finfun"><span class="id" title="abbreviation">finfun</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="eq_dffun"><span class="id" title="lemma">eq_dffun</span></a> (<span class="id" title="var">g1</span> <span class="id" title="var">g2</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.finfun.html#DepPlainTheory.rT"><span class="id" title="variable">rT</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a>) :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.finfun.html#g1"><span class="id" title="variable">g1</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#g2"><span class="id" title="variable">g2</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#finfun"><span class="id" title="abbreviation">finfun</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#g1"><span class="id" title="variable">g1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#finfun"><span class="id" title="abbreviation">finfun</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#g2"><span class="id" title="variable">g2</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="total_fun"><span class="id" title="definition">total_fun</span></a> <span class="id" title="var">g</span> <span class="id" title="var">x</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#Tagged"><span class="id" title="definition">Tagged</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#DepPlainTheory.rT"><span class="id" title="variable">rT</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#DepPlainTheory.rT"><span class="id" title="variable">rT</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a>).<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="tfgraph"><span class="id" title="definition">tfgraph</span></a> <span class="id" title="var">f</span> := <a class="idref" href="mathcomp.ssreflect.tuple.html#codom_tuple"><span class="id" title="definition">codom_tuple</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#total_fun"><span class="id" title="definition">total_fun</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="codom_tffun"><span class="id" title="lemma">codom_tffun</span></a> <span class="id" title="var">f</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#codom"><span class="id" title="definition">codom</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#total_fun"><span class="id" title="definition">total_fun</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#tfgraph"><span class="id" title="definition">tfgraph</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a>. <br/>
<br/>
<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="tfgraph_inj"><span class="id" title="lemma">tfgraph_inj</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#tfgraph"><span class="id" title="definition">tfgraph</span></a>. <br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="family_mem"><span class="id" title="definition">family_mem</span></a> <span class="id" title="var">mF</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#fbeb549dcb6350fb8ceb1bda39acce60"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#fbeb549dcb6350fb8ceb1bda39acce60"><span class="id" title="notation">pred</span></a> <span class="id" title="var">f</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#fbeb549dcb6350fb8ceb1bda39acce60"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#fT"><span class="id" title="abbreviation">fT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#fbeb549dcb6350fb8ceb1bda39acce60"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#ce8c9a990e3e773a56ef37417d3761c6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#ce8c9a990e3e773a56ef37417d3761c6"><span class="id" title="notation">∀</span></a> <span class="id" title="var">x</span><a class="idref" href="mathcomp.ssreflect.fintype.html#f3be25edeb0349b0a76405eded9d0b98"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#in_mem"><span class="id" title="definition">in_mem</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.finfun.html#mF"><span class="id" title="variable">mF</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a>)<a class="idref" href="mathcomp.ssreflect.fintype.html#ce8c9a990e3e773a56ef37417d3761c6"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#fbeb549dcb6350fb8ceb1bda39acce60"><span class="id" title="notation">]</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variables</span> (<a name="DepPlainTheory.pT"><span class="id" title="variable">pT</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#predType"><span class="id" title="record">predType</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#DepPlainTheory.rT"><span class="id" title="variable">rT</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a>)) (<a name="DepPlainTheory.F"><span class="id" title="variable">F</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.finfun.html#pT"><span class="id" title="variable">pT</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a>).<br/>
<br/>
</div>
<div class="doc">
Helper for defining notation for function families.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Lemma</span> <a name="familyP"><span class="id" title="lemma">familyP</span></a> <span class="id" title="var">f</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#DepPlainTheory.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#family_mem"><span class="id" title="definition">family_mem</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#fmem"><span class="id" title="definition">fmem</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#DepPlainTheory.F"><span class="id" title="variable">F</span></a>)).<br/>
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<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.finfun.html#DepPlainTheory"><span class="id" title="section">DepPlainTheory</span></a>.<br/>
<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="family"><span class="id" title="abbreviation">family</span></a> <span class="id" title="var">F</span> := (<a class="idref" href="mathcomp.ssreflect.finfun.html#family_mem"><span class="id" title="definition">family_mem</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#fmem"><span class="id" title="definition">fmem</span></a> <span class="id" title="var">F</span>)).<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="InheritedStructures"><span class="id" title="section">InheritedStructures</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variable</span> <a name="InheritedStructures.aT"><span class="id" title="variable">aT</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>.<br/>
<span class="id" title="keyword">Notation</span> <a name="dffun_aT"><span class="id" title="abbreviation">dffun_aT</span></a> <span class="id" title="var">rT</span> <span class="id" title="var">rS</span> := <a class="idref" href="mathcomp.ssreflect.finfun.html#15e3a46a8a60865a7b5408d87034cd4e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#15e3a46a8a60865a7b5408d87034cd4e"><span class="id" title="notation">dffun</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.finfun.html#InheritedStructures.aT"><span class="id" title="variable">aT</span></a>, <span class="id" title="var">rT</span> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <span class="id" title="var">rS</span><a class="idref" href="mathcomp.ssreflect.finfun.html#15e3a46a8a60865a7b5408d87034cd4e"><span class="id" title="notation">}</span></a>.<br/>
<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">finfun_eqType</span> (<span class="id" title="var">rT</span> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) :=<br/>
<a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.EqType"><span class="id" title="abbreviation">EqType</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">ffun</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#InheritedStructures.aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">}</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#eqMixin"><span class="id" title="lemma">eqMixin</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#94502d422d70bbaf4f390946d9885b7c"><span class="id" title="notation">fun</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#94502d422d70bbaf4f390946d9885b7c"><span class="id" title="notation">⇒</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#rT"><span class="id" title="variable">rT</span></a>)).<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">dfinfun_eqType</span> <span class="id" title="var">rT</span> :=<br/>
<a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.EqType"><span class="id" title="abbreviation">EqType</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#dffun_aT"><span class="id" title="abbreviation">dffun_aT</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#rT"><span class="id" title="variable">rT</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) (<a class="idref" href="mathcomp.ssreflect.finfun.html#eqMixin"><span class="id" title="lemma">eqMixin</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#rT"><span class="id" title="variable">rT</span></a>).<br/>
<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">finfun_choiceType</span> (<span class="id" title="var">rT</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Exports.choiceType"><span class="id" title="abbreviation">choiceType</span></a>) :=<br/>
<a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Exports.ChoiceType"><span class="id" title="abbreviation">ChoiceType</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">ffun</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#InheritedStructures.aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">}</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#choiceMixin"><span class="id" title="lemma">choiceMixin</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#94502d422d70bbaf4f390946d9885b7c"><span class="id" title="notation">fun</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#94502d422d70bbaf4f390946d9885b7c"><span class="id" title="notation">⇒</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#rT"><span class="id" title="variable">rT</span></a>)).<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">dfinfun_choiceType</span> <span class="id" title="var">rT</span> :=<br/>
<a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Exports.ChoiceType"><span class="id" title="abbreviation">ChoiceType</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#dffun_aT"><span class="id" title="abbreviation">dffun_aT</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#rT"><span class="id" title="variable">rT</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Exports.choiceType"><span class="id" title="abbreviation">choiceType</span></a>) (<a class="idref" href="mathcomp.ssreflect.finfun.html#choiceMixin"><span class="id" title="lemma">choiceMixin</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#rT"><span class="id" title="variable">rT</span></a>).<br/>
<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">finfun_countType</span> (<span class="id" title="var">rT</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.Exports.countType"><span class="id" title="abbreviation">countType</span></a>) :=<br/>
<a class="idref" href="mathcomp.ssreflect.choice.html#Countable.Exports.CountType"><span class="id" title="abbreviation">CountType</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">ffun</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#InheritedStructures.aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">}</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#countMixin"><span class="id" title="lemma">countMixin</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#94502d422d70bbaf4f390946d9885b7c"><span class="id" title="notation">fun</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#94502d422d70bbaf4f390946d9885b7c"><span class="id" title="notation">⇒</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#rT"><span class="id" title="variable">rT</span></a>)).<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">dfinfun_countType</span> <span class="id" title="var">rT</span> :=<br/>
<a class="idref" href="mathcomp.ssreflect.choice.html#Countable.Exports.CountType"><span class="id" title="abbreviation">CountType</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#dffun_aT"><span class="id" title="abbreviation">dffun_aT</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#rT"><span class="id" title="variable">rT</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.Exports.countType"><span class="id" title="abbreviation">countType</span></a>) (<a class="idref" href="mathcomp.ssreflect.finfun.html#countMixin"><span class="id" title="lemma">countMixin</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#rT"><span class="id" title="variable">rT</span></a>).<br/>
<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">finfun_finType</span> (<span class="id" title="var">rT</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) :=<br/>
<a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.FinType"><span class="id" title="abbreviation">FinType</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">ffun</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#InheritedStructures.aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">}</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#finMixin"><span class="id" title="definition">finMixin</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#94502d422d70bbaf4f390946d9885b7c"><span class="id" title="notation">fun</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#94502d422d70bbaf4f390946d9885b7c"><span class="id" title="notation">⇒</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#rT"><span class="id" title="variable">rT</span></a>)).<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">dfinfun_finType</span> <span class="id" title="var">rT</span> :=<br/>
<a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.FinType"><span class="id" title="abbreviation">FinType</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#dffun_aT"><span class="id" title="abbreviation">dffun_aT</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#rT"><span class="id" title="variable">rT</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<a class="idref" href="mathcomp.ssreflect.finfun.html#finMixin"><span class="id" title="definition">finMixin</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#rT"><span class="id" title="variable">rT</span></a>).<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.finfun.html#InheritedStructures"><span class="id" title="section">InheritedStructures</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="FunPlainTheory"><span class="id" title="section">FunPlainTheory</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variables</span> (<a name="FunPlainTheory.aT"><span class="id" title="variable">aT</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<a name="FunPlainTheory.rT"><span class="id" title="variable">rT</span></a> : <span class="id" title="keyword">Type</span>).<br/>
<span class="id" title="keyword">Notation</span> <a name="fT"><span class="id" title="abbreviation">fT</span></a> := <a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">ffun</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#FunPlainTheory.aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#FunPlainTheory.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">}</span></a>.<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.ssreflect.finfun.html#fT"><span class="id" title="abbreviation">fT</span></a>) (<span class="id" title="var">R</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#FunPlainTheory.rT"><span class="id" title="variable">rT</span></a>).<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="fgraph"><span class="id" title="definition">fgraph</span></a> <span class="id" title="var">f</span> := <a class="idref" href="mathcomp.ssreflect.tuple.html#codom_tuple"><span class="id" title="definition">codom_tuple</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="Finfun"><span class="id" title="definition">Finfun</span></a> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#FunPlainTheory.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.ssreflect.tuple.html#c3913abe839346eb60d82da74b0b1f67"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.ssreflect.tuple.html#c3913abe839346eb60d82da74b0b1f67"><span class="id" title="notation">tuple</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#FunPlainTheory.rT"><span class="id" title="variable">rT</span></a>) := <a class="idref" href="mathcomp.ssreflect.finfun.html#486743bb05c6aa8b9d64fd3cec29ee79"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#486743bb05c6aa8b9d64fd3cec29ee79"><span class="id" title="notation">ffun</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.finfun.html#486743bb05c6aa8b9d64fd3cec29ee79"><span class="id" title="notation">⇒</span></a> <a class="idref" href="mathcomp.ssreflect.tuple.html#tnth"><span class="id" title="definition">tnth</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#G"><span class="id" title="variable">G</span></a> (<a class="idref" href="mathcomp.ssreflect.fintype.html#enum_rank"><span class="id" title="definition">enum_rank</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a>)<a class="idref" href="mathcomp.ssreflect.finfun.html#486743bb05c6aa8b9d64fd3cec29ee79"><span class="id" title="notation">]</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="tnth_fgraph"><span class="id" title="lemma">tnth_fgraph</span></a> <span class="id" title="var">f</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.ssreflect.tuple.html#tnth"><span class="id" title="definition">tnth</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#fgraph"><span class="id" title="definition">fgraph</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a>) <a class="idref" href="mathcomp.ssreflect.finfun.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.ssreflect.fintype.html#enum_val"><span class="id" title="definition">enum_val</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#i"><span class="id" title="variable">i</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="FinfunK"><span class="id" title="lemma">FinfunK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#Finfun"><span class="id" title="definition">Finfun</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#fgraph"><span class="id" title="definition">fgraph</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="fgraphK"><span class="id" title="lemma">fgraphK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#fgraph"><span class="id" title="definition">fgraph</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#Finfun"><span class="id" title="definition">Finfun</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="fgraph_ffun0"><span class="id" title="lemma">fgraph_ffun0</span></a> <span class="id" title="var">aT0</span> : <a class="idref" href="mathcomp.ssreflect.finfun.html#fgraph"><span class="id" title="definition">fgraph</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#ffun0"><span class="id" title="lemma">ffun0</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#aT0"><span class="id" title="variable">aT0</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#b8b2ebc8e1a8b9aa935c0702efb5dccf"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#b8b2ebc8e1a8b9aa935c0702efb5dccf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#FunPlainTheory.rT"><span class="id" title="variable">rT</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="codom_ffun"><span class="id" title="lemma">codom_ffun</span></a> <span class="id" title="var">f</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#codom"><span class="id" title="definition">codom</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#fgraph"><span class="id" title="definition">fgraph</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a>. <br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="tagged_tfgraph"><span class="id" title="lemma">tagged_tfgraph</span></a> <span class="id" title="var">f</span> : @<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.ssreflect.finfun.html#FunPlainTheory.rT"><span class="id" title="variable">rT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#tagged"><span class="id" title="definition">tagged</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#tfgraph"><span class="id" title="definition">tfgraph</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#fgraph"><span class="id" title="definition">fgraph</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="eq_ffun"><span class="id" title="lemma">eq_ffun</span></a> (<span class="id" title="var">g1</span> <span class="id" title="var">g2</span> : <a class="idref" href="mathcomp.ssreflect.finfun.html#FunPlainTheory.aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#FunPlainTheory.rT"><span class="id" title="variable">rT</span></a>) : <a class="idref" href="mathcomp.ssreflect.finfun.html#g1"><span class="id" title="variable">g1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#g2"><span class="id" title="variable">g2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#finfun"><span class="id" title="abbreviation">finfun</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#g1"><span class="id" title="variable">g1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#finfun"><span class="id" title="abbreviation">finfun</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#g2"><span class="id" title="variable">g2</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="fgraph_codom"><span class="id" title="lemma">fgraph_codom</span></a> <span class="id" title="var">f</span> : <a class="idref" href="mathcomp.ssreflect.finfun.html#fgraph"><span class="id" title="definition">fgraph</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.tuple.html#codom_tuple"><span class="id" title="definition">codom_tuple</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="ffun_on_mem"><span class="id" title="definition">ffun_on_mem</span></a> (<span class="id" title="var">mR</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#mem_pred"><span class="id" title="inductive">mem_pred</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#FunPlainTheory.rT"><span class="id" title="variable">rT</span></a>) := <a class="idref" href="mathcomp.ssreflect.finfun.html#family_mem"><span class="id" title="definition">family_mem</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.ssreflect.finfun.html#FunPlainTheory.aT"><span class="id" title="variable">aT</span></a> ⇒ <a class="idref" href="mathcomp.ssreflect.finfun.html#mR"><span class="id" title="variable">mR</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="ffun_onP"><span class="id" title="lemma">ffun_onP</span></a> <span class="id" title="var">R</span> <span class="id" title="var">f</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#R"><span class="id" title="variable">R</span></a>) (<a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#ffun_on_mem"><span class="id" title="definition">ffun_on_mem</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#R"><span class="id" title="variable">R</span></a>)).<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.finfun.html#FunPlainTheory"><span class="id" title="section">FunPlainTheory</span></a>.<br/>
<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="ffun_on"><span class="id" title="abbreviation">ffun_on</span></a> <span class="id" title="var">R</span> := (<a class="idref" href="mathcomp.ssreflect.finfun.html#ffun_on_mem"><span class="id" title="definition">ffun_on_mem</span></a> <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">R</span>)).<br/>
<span class="id" title="keyword">Notation</span> <a name="445fe95595384a6d39644c554d9b4997"><span class="id" title="notation">"</span></a>@ 'ffun_on' aT R" :=<br/>
(<a class="idref" href="mathcomp.ssreflect.finfun.html#ffun_on"><span class="id" title="abbreviation">ffun_on</span></a> <span class="id" title="var">R</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#simpl_pred"><span class="id" title="definition">simpl_pred</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#finfun_of"><span class="id" title="inductive">finfun_of</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#Phant"><span class="id" title="constructor">Phant</span></a> (<span class="id" title="var">aT</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a> <span class="id" title="var">_</span>))))<br/>
(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 10, <span class="id" title="var">aT</span>, <span class="id" title="var">R</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 9).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="nth_fgraph_ord"><span class="id" title="lemma">nth_fgraph_ord</span></a> <span class="id" title="var">T</span> <span class="id" title="var">n</span> (<span class="id" title="var">x0</span> : <a class="idref" href="mathcomp.ssreflect.finfun.html#T"><span class="id" title="variable">T</span></a>) (<span class="id" title="var">i</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#545d9d6249a673300f950a2a8b8a930b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#545d9d6249a673300f950a2a8b8a930b"><span class="id" title="notation">I_n</span></a>) <span class="id" title="var">f</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x0"><span class="id" title="variable">x0</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#fgraph"><span class="id" title="definition">fgraph</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a>) <a class="idref" href="mathcomp.ssreflect.finfun.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#i"><span class="id" title="variable">i</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="Support"><span class="id" title="section">Support</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variables</span> (<a name="Support.aT"><span class="id" title="variable">aT</span></a> : <span class="id" title="keyword">Type</span>) (<a name="Support.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>).<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="support_for"><span class="id" title="definition">support_for</span></a> <span class="id" title="var">y</span> (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.ssreflect.finfun.html#Support.aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#Support.rT"><span class="id" title="variable">rT</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">pred</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">]</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="supportE"><span class="id" title="lemma">supportE</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">f</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#support_for"><span class="id" title="definition">support_for</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>. <br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.finfun.html#Support"><span class="id" title="section">Support</span></a>.<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="698544468b858f103778b531f3023430"><span class="id" title="notation">"</span></a>y .-support" := (<a class="idref" href="mathcomp.ssreflect.finfun.html#support_for"><span class="id" title="definition">support_for</span></a> <span class="id" title="var">y</span>)<br/>
(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "y .-support") : <span class="id" title="var">fun_scope</span>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="EqTheory"><span class="id" title="section">EqTheory</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variables</span> (<a name="EqTheory.aT"><span class="id" title="variable">aT</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<a name="EqTheory.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>).<br/>
<span class="id" title="keyword">Notation</span> <a name="fT"><span class="id" title="abbreviation">fT</span></a> := <a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">ffun</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#EqTheory.aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#EqTheory.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">}</span></a>.<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">y</span> : <a class="idref" href="mathcomp.ssreflect.finfun.html#EqTheory.rT"><span class="id" title="variable">rT</span></a>) (<span class="id" title="var">D</span> : <a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">pred</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#EqTheory.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">pred</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#EqTheory.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.ssreflect.finfun.html#fT"><span class="id" title="abbreviation">fT</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="supportP"><span class="id" title="lemma">supportP</span></a> <span class="id" title="var">y</span> <span class="id" title="var">D</span> <span class="id" title="var">g</span> :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.ssreflect.finfun.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#698544468b858f103778b531f3023430"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#698544468b858f103778b531f3023430"><span class="id" title="notation">support</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#D"><span class="id" title="variable">D</span></a>).<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="pfamily_mem"><span class="id" title="definition">pfamily_mem</span></a> <span class="id" title="var">y</span> <span class="id" title="var">mD</span> (<span class="id" title="var">mF</span> : <a class="idref" href="mathcomp.ssreflect.finfun.html#EqTheory.aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#mem_pred"><span class="id" title="inductive">mem_pred</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#EqTheory.rT"><span class="id" title="variable">rT</span></a>) :=<br/>
<a class="idref" href="mathcomp.ssreflect.finfun.html#family"><span class="id" title="abbreviation">family</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.ssreflect.finfun.html#EqTheory.aT"><span class="id" title="variable">aT</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#in_mem"><span class="id" title="definition">in_mem</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#mD"><span class="id" title="variable">mD</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred_of_simpl"><span class="id" title="definition">pred_of_simpl</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#mF"><span class="id" title="variable">mF</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#pred1"><span class="id" title="definition">pred1</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#y"><span class="id" title="variable">y</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="pfamilyP"><span class="id" title="lemma">pfamilyP</span></a> (<span class="id" title="var">pT</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#predType"><span class="id" title="record">predType</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#EqTheory.rT"><span class="id" title="variable">rT</span></a>) <span class="id" title="var">y</span> <span class="id" title="var">D</span> (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.ssreflect.finfun.html#EqTheory.aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#pT"><span class="id" title="variable">pT</span></a>) <span class="id" title="var">f</span> :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#698544468b858f103778b531f3023430"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#698544468b858f103778b531f3023430"><span class="id" title="notation">support</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#D"><span class="id" title="variable">D</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a>)<br/>
(<a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#pfamily_mem"><span class="id" title="definition">pfamily_mem</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#D"><span class="id" title="variable">D</span></a>) (<a class="idref" href="mathcomp.ssreflect.finfun.html#fmem"><span class="id" title="definition">fmem</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#F"><span class="id" title="variable">F</span></a>)).<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="pffun_on_mem"><span class="id" title="definition">pffun_on_mem</span></a> <span class="id" title="var">y</span> <span class="id" title="var">mD</span> <span class="id" title="var">mR</span> := <a class="idref" href="mathcomp.ssreflect.finfun.html#pfamily_mem"><span class="id" title="definition">pfamily_mem</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#mD"><span class="id" title="variable">mD</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="mathcomp.ssreflect.finfun.html#mR"><span class="id" title="variable">mR</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="pffun_onP"><span class="id" title="lemma">pffun_onP</span></a> <span class="id" title="var">y</span> <span class="id" title="var">D</span> <span class="id" title="var">R</span> <span class="id" title="var">f</span> :<br/>
<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#698544468b858f103778b531f3023430"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#698544468b858f103778b531f3023430"><span class="id" title="notation">support</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#image"><span class="id" title="abbreviation">image</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">}</span></a>)<br/>
(<a class="idref" href="mathcomp.ssreflect.finfun.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#pffun_on_mem"><span class="id" title="definition">pffun_on_mem</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#D"><span class="id" title="variable">D</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#R"><span class="id" title="variable">R</span></a>)).<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.finfun.html#EqTheory"><span class="id" title="section">EqTheory</span></a>.<br/>
<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="pfamily"><span class="id" title="abbreviation">pfamily</span></a> <span class="id" title="var">y</span> <span class="id" title="var">D</span> <span class="id" title="var">F</span> := (<a class="idref" href="mathcomp.ssreflect.finfun.html#pfamily_mem"><span class="id" title="definition">pfamily_mem</span></a> <span class="id" title="var">y</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">D</span>) (<a class="idref" href="mathcomp.ssreflect.finfun.html#fmem"><span class="id" title="definition">fmem</span></a> <span class="id" title="var">F</span>)).<br/>
<span class="id" title="keyword">Notation</span> <a name="pffun_on"><span class="id" title="abbreviation">pffun_on</span></a> <span class="id" title="var">y</span> <span class="id" title="var">D</span> <span class="id" title="var">R</span> := (<a class="idref" href="mathcomp.ssreflect.finfun.html#pffun_on_mem"><span class="id" title="definition">pffun_on_mem</span></a> <span class="id" title="var">y</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">D</span>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">R</span>)).<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="FinDepTheory"><span class="id" title="section">FinDepTheory</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variables</span> (<a name="FinDepTheory.aT"><span class="id" title="variable">aT</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<a name="FinDepTheory.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.ssreflect.finfun.html#aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>).<br/>
<span class="id" title="keyword">Notation</span> <a name="fT"><span class="id" title="abbreviation">fT</span></a> := <a class="idref" href="mathcomp.ssreflect.finfun.html#15e3a46a8a60865a7b5408d87034cd4e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#15e3a46a8a60865a7b5408d87034cd4e"><span class="id" title="notation">dffun</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.finfun.html#FinDepTheory.aT"><span class="id" title="variable">aT</span></a>, <a class="idref" href="mathcomp.ssreflect.finfun.html#FinDepTheory.rT"><span class="id" title="variable">rT</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#15e3a46a8a60865a7b5408d87034cd4e"><span class="id" title="notation">}</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="card_family"><span class="id" title="lemma">card_family</span></a> (<span class="id" title="var">F</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#FinDepTheory.rT"><span class="id" title="variable">rT</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a>)) :<br/>
<a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|(</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#family"><span class="id" title="abbreviation">family</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#simpl_pred"><span class="id" title="definition">simpl_pred</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#fT"><span class="id" title="abbreviation">fT</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">)|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#foldr"><span class="id" title="definition">foldr</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#muln"><span class="id" title="definition">muln</span></a> 1 <a class="idref" href="mathcomp.ssreflect.fintype.html#2ea807d068496425ebd93f2c454c8460"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#2ea807d068496425ebd93f2c454c8460"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#2ea807d068496425ebd93f2c454c8460"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#2ea807d068496425ebd93f2c454c8460"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#FinDepTheory.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#2ea807d068496425ebd93f2c454c8460"><span class="id" title="notation">]</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="card_dep_ffun"><span class="id" title="lemma">card_dep_ffun</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#fT"><span class="id" title="abbreviation">fT</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#foldr"><span class="id" title="definition">foldr</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#muln"><span class="id" title="definition">muln</span></a> 1 <a class="idref" href="mathcomp.ssreflect.fintype.html#2ea807d068496425ebd93f2c454c8460"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#2ea807d068496425ebd93f2c454c8460"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#FinDepTheory.rT"><span class="id" title="variable">rT</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#2ea807d068496425ebd93f2c454c8460"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#2ea807d068496425ebd93f2c454c8460"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#FinDepTheory.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#2ea807d068496425ebd93f2c454c8460"><span class="id" title="notation">]</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.finfun.html#FinDepTheory"><span class="id" title="section">FinDepTheory</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="FinFunTheory"><span class="id" title="section">FinFunTheory</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variables</span> <a name="FinFunTheory.aT"><span class="id" title="variable">aT</span></a> <a name="FinFunTheory.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>.<br/>
<span class="id" title="keyword">Notation</span> <a name="fT"><span class="id" title="abbreviation">fT</span></a> := <a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">ffun</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#FinFunTheory.aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#FinFunTheory.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">}</span></a>.<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">D</span> : <a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">pred</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#FinFunTheory.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">pred</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#FinFunTheory.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.ssreflect.finfun.html#FinFunTheory.aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#FinFunTheory.rT"><span class="id" title="variable">rT</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="card_pfamily"><span class="id" title="lemma">card_pfamily</span></a> <span class="id" title="var">y0</span> <span class="id" title="var">D</span> <span class="id" title="var">F</span> :<br/>
<a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#pfamily"><span class="id" title="abbreviation">pfamily</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#y0"><span class="id" title="variable">y0</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#foldr"><span class="id" title="definition">foldr</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#muln"><span class="id" title="definition">muln</span></a> 1 <a class="idref" href="mathcomp.ssreflect.fintype.html#c3ff8d84d4e3e273bebfcf7502deb41a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#c3ff8d84d4e3e273bebfcf7502deb41a"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#c3ff8d84d4e3e273bebfcf7502deb41a"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#c3ff8d84d4e3e273bebfcf7502deb41a"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#D"><span class="id" title="variable">D</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#c3ff8d84d4e3e273bebfcf7502deb41a"><span class="id" title="notation">]</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="card_pffun_on"><span class="id" title="lemma">card_pffun_on</span></a> <span class="id" title="var">y0</span> <span class="id" title="var">D</span> <span class="id" title="var">R</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#pffun_on"><span class="id" title="abbreviation">pffun_on</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#y0"><span class="id" title="variable">y0</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#D"><span class="id" title="variable">D</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="card_ffun_on"><span class="id" title="lemma">card_ffun_on</span></a> <span class="id" title="var">R</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#445fe95595384a6d39644c554d9b4997"><span class="id" title="notation">@</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#445fe95595384a6d39644c554d9b4997"><span class="id" title="notation">ffun_on</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#FinFunTheory.aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#FinFunTheory.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="card_ffun"><span class="id" title="lemma">card_ffun</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#fT"><span class="id" title="abbreviation">fT</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#FinFunTheory.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#FinFunTheory.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.finfun.html#FinFunTheory"><span class="id" title="section">FinFunTheory</span></a>.<br/>
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